Gompertz Distribution Research Articles

Gompertz-Weibull {Gompertz} Type I: A Four-Parameter Generalization of the Weibull Family

AbstractIn this paper, we provide a construction method that provides greater flexibility in modeling real-world phenomenon by generalizing univariate probability distributions. The Gompertz distribution is used to provide new classes of models through T-X{Y} construction, in a manner we will define as a “proper” generalization, illustrated by application to the popular Weibull family. We provide moments and other properties of the model, including the necessary equations for maximum likelihood estimation, and we fit a set of real data to the model. A framework for classifying generalizations that includes many previously discovered methods and considerations for the research infrastructure are proposed in our methods and discussion.

A defective cure rate quantile regression model for male breast cancer data

In this article, we particularly address the problem of assessing the impact of different prognostic factors, such as clinical stage and age, on the specific survival times of men with breast cancer when cure is a possibility. To this end, we developed a quantile regression model for survival data in the presence of long-term survivors based on the generalized Gompertz distribution in a defective version, which is conveniently reparametrized in terms of the q-th quantile and then linked to covariates via a logarithm link function. This proposal allows us to obtain how each variable affects the survival times in different quantiles. In addition, we are able to study the effects of covariates on the cure rate as well. We consider Markov Chain Monte Carlo methods to develop a Bayesian analysis in the proposed model and we evaluate its performance through Monte Carlo simulation studies. Finally, we illustrate the application of our model in a data set about male breast cancer from Brazil analyzed for the very first time.

Digitized fuel load survey in commercial and university office buildings for fire safety assessment

Fuel load significantly affects fire development in compartments, and its design value derives from the statistical results of numerous surveys. This work enhances the fuel load database by employing a digitized survey method to assess 27 university and commercial offices in Hong Kong and Mainland, China. The results revealed significant differences in fuel load densities: university offices averaged 382 MJ/m2 (SD: 297 MJ/m2) and commercial offices averaged 1804 MJ/m2 (SD: 1319 MJ/m2). Moreover, it proposed an online questionnaire method to overcome onsite access limitations. Furthermore, it identified higher fuel load densities in commercial offices due to higher paper-made content and greater occupancy density than university offices. The fuel load of university offices was lower than that in previous surveys and design codes, while commercial fuel load was higher. Notably, there is an increasing tendency of fuel load density and plastic combustible composition over the years. Additionally, it considers that Gompertz distribution better fits cumulative probabilities of fuel load density data with a small sample size. Overall, it provides a valuable database for future fire scenario design, fire codes edition, and fire safety assessment and discusses future collaboration with AI applications.

Analyzing Cancer Survival Times Using the Exponentiated Fréchet-Gompertz Distribution

Understanding the survival of cancer patients is essential for determining optimal treatment strategies. This research introduces a robust distribution for analyzing survival data known as the Exponential Fr ́echet-Gompertz distribution (EFG). The EFG combines the unique characteristics of the Exponential Fr ́echet and Gompertz distributions, enhancing its efficiency and providing greater flexibility in representing complex datasets. The study specifically investigates the EFG distribution’s effectiveness in modeling cancer patients’ survival times. A comprehensive analysis of the distribution’s properties is presented, including the quantile and quartile functions, shape indices, moments, moment-generating function, characteristic function, mean residual life, mean waiting time, R ́enyi entropy, and order statistics. The parameters of the distribution are derivedusing five distinct methods: Maximum Likelihood Estimation (MLE), Ordinary Least Squares (OLS), Weighted Least Squares (WLS), Cram ́er-von Mises (CVM), and Maximum Product of Spacings (MPS). A Monte Carlo simulation technique is employed to evaluate the performance of these estimation methods. The simulation results indicate that as sample size increases, the meansquare error (MSE) values for all estimators decrease. Notably, the MLE exhibits the lowest MSE, while the MPS has the highest MSE, particularly for smaller sample sizes. Furthermore, the study presents a comprehensive comparison of the effectiveness of these estimation methods in analyzing survival times for various cancer types, including bladder, bone, blood and brain cancer. The results indicate that the EFG distribution is an optimal model for representing the survival times of patients across these different cancers data. Furthermore, the W.LS method yielded superior estimations for most datasets concerning cancer patient survival. Overall, the EFG distribution demonstrates exceptional capability in accurately fitting survival times for cancer patients compared to other competing distributions.

Determinants of Long COVID illness duration and service use-an Irish retrospective cohort study

Abstract The Follow-up After Disease Acquisition (FADA) study is a retrospective cohort study of COVID-19 recovery, aimed to characterise the nature and severity of Long COVID (LC) in an Irish community-based sample. Duration analysis is conducted applying Cox’s proportional hazards model to those self-reported recovered from LC (n = 577) and a Gompertz distribution parametric model, determined best fit by AIC, for those still ill with LC (n = 928) at time of survey completion. A negative binomial model is estimated to understand the variation of services accessed for LC. The Hazard in this instance is recovery(<1 = contributing factor to duration). Factors significantly contributing to duration of illness are never married (Hazard Ratio (HR):0.689;p:0.01) higher physical activity levels prior to first C-19 infection (HR:0.964; p:0.06) and number of COVID-19 infections (H:0.926;p:0.07). Those that contract more recent variants are likely to recover quicker from their LC illness (baseline Wildtype. Alpha HR:1.7;p:0.0; Delta HR:3.36;p:0.0; Omicron HR:3.8; p:0.0). In variation of services used for LC, there are some major differences between those self-reporting as recovered from LC at time of survey completion and those still ill. Of those recovered, men, those sickest with LC and sick longest use a wider variation of services. Single people use a significantly narrower range of health services to combat their illness. For those still ill higher education, higher levels of physical activity pre COVID-19, healthcare workers, higher rates of comorbidity, those sick longest and those whose health has been most impacted by LC use a wider variation of services. A poorer health rating negatively impacts variation of services accessed for LC illness. More recent strains of COVID-19 are significantly less likely to lead to LC, this offers hope for the future, albeit more research is required to investigate factors to improve recovery rates for those already experiencing LC. Key messages • Those with Long COVID that recover quickly are fundamentally different to those that do not. • The number of COVID-19 infections is a key indicator of length of Long COVID illness, we need to protect our most vulnerable and put appropriate safeguards in place for healthcare provision.

Order-restricted statistical inference and optimal censoring scheme for Gompertz distribution based on joint type-II progressive censored data

This paper considers the order-restricted statistical inference for two populations based on the joint type-II progressive censoring scheme. The lifetime distributions of the two populations are supposed to follow the Gompertz distribution with the same shape parameter but different scale parameters. The maximum likelihood estimates of the unknown parameters are derived by employing the Newton-Raphson algorithm and the expectation-maximization algorithm, respectively. The Fisher information matrix is then employed to construct asymptotic confidence intervals. For Bayes estimation, we assume an ordered Beta-Gamma prior for the scale parameters and a Gamma prior for the common shape parameter. Bayes estimations and the highest posterior density credible intervals for unknown parameters under two different loss functions are obtained with the importance sampling technique. To evaluate the performance of order-restricted inference, extensive Monte Carlo simulations are performed and two air-conditioning systems datasets are used to illustrate the proposed inference methods. In addition, the results are compared with the case when there is no order restriction on the parameters. Finally, the optimal censoring scheme is obtained by four optimality criteria.

The entropy-transformed Gompertz distribution: Distributional insights and cross-disciplinary utilizations

A novel two-parameter continuous model titled the entropy-transformed Gompertz (ETGPZ) distribution has been developed via the entropy transform. A new framework has been investigated and found to meet the criteria of the probability function. By significantly improving the functional shape and having the ability to model the most likely form of the hazard rate function, this new modification has increased the adaptability of the typical distribution. Some of its core characteristics, such as its statistical and computational features, are clearly presented. A thorough simulation analysis has been done to examine the final behavior of maximum likelihood estimators while estimating model parameters. We assess the performance and practical applicability of the ETGPZ distribution using eight real datasets from engineering and biomedical fields. The results demonstrate that the ETGPZ outperforms the baseline Gompertz (GPZ) distribution, highlighting its superiority and broader potential for various applications.

Modeling Chronic Disease Mortality by Methods From Accelerated Life Testing.

We propose a parametric model for describing chronic disease mortality from cohort data and illustrate its use for Type 2 diabetes. The model uses ideas from accelerated life testing in reliability theory and conceptualizes the occurrence of a chronic disease as putting the observational unit to an enhanced stress level, which is supposed to shorten its lifetime. It further addresses the issue of semi-competing risk, that is, the asymmetry of death and diagnosis of disease, where the disease can be diagnosed before death, but not after. With respect to the cohort structure of the data, late entry into the cohort is taken into account and prevalent as well as incident cases inform the analysis. We finally give an extension of the model that allows age at disease diagnosis to be observed not exactly, but only partially within an interval. Model parameters can be straightforwardly estimated by Maximum Likelihood, using the assumption of a Gompertz distribution we show in a small simulation study that this works well. Data of the Cardiovascular Disease, Living and Ageing in Halle (CARLA) study, a population-based cohort in the city of Halle (Saale) in the eastern part of Germany, are used for illustration.

Prognostic factors for premature cardiovascular disease mortality in Malaysia: a modelling approach using semi-parametric and parametric survival analysis with national health and morbidity survey linked mortality data.

Cardiovascular disease (CVD) is the leading cause of premature mortality worldwide. Despite existing research on CVD risk factors, the study of premature CVD mortality in Malaysia remains limited. This study employs survival analysis to model modifiable risk factors associated with premature CVD mortality among Malaysian adults. We utilised data from Malaysia's National Health and Morbidity Survey (NHMS) conducted in 2006, 2011, and 2015, linked with mortality records. The cohort comprised individuals aged 18 to 70 during the NHMS interview. Follow-up extended to 2021, focusing on CVD-related premature mortality between ages 30 and 70. We employed six survival models: a semi-parametric Cox proportional hazard (PH) and five parametric survival models, which were Exponential, Weibull, Gompertz, log-normal and log-logistic distributions using R software. The age standardized incidence rate (ASIR) of premature CVD mortality was calculated per 1000 person-years. Among 63,722 participants, 886 (1.4%) experienced premature CVD mortality, with an ASIR of 1.80 per 1000 person-years. The best-fit models (based on AIC value) were the stratified Cox model by age (semi-parametric) and the log-normal accelerated failure time (AFT) model (parametric). Males had higher risk (Hazard Ratio, HR = 2.68) and experienced 49% shorter survival time (Event Time Ratio, ETR = 0.51) compared to females. Compared to Chinese ethnicity, Indians, Malays, and other Bumiputera had higher HR and lower survival times. Rural residents and those with lower education also faced increased HRs and reduced survival times. Diabetes (diagnosed: HR = 3.26, ETR = 0.37; undiagnosed: HR = 1.63, ETR = 0.63), hypertension (diagnosed: HR = 1.84, ETR = 0.53; undiagnosed: HR = 1.46, ETR = 0.68), and undiagnosed hypercholesterolemia (HR = 1.31, ETR = 0.80) increased risk and decreased survival times. Additionally, current smoking and abdominal obesity elevated risk (HR = 1.38, 1.60) and shortened survival (ETR = 0.81, 0.71). The semi-parametric and parametric survival models both highlight the considerable impact of socioeconomic status and modifiable risk factors on premature CVD mortality, underscoring the imperative for targeted interventions to effectively mitigate these effects.

About the Residual Time of Human Life

Many researchers believe that human lifetime in the interval from 20 to 80 years is approximately described by the Gompertz distribution. It is natural to consider the problem of the distribution of the residual human lifetime at the final stage, that is, after 80 years. To do this, in this study, a statistical analysis of some statistical models on real data is carried out. Several models are presented, which, according to the authors, can adequately describe data on mortality after 80 years.

Optimizing Inventory Model for Negative Exponential and Linear Split Demand and Gompertz Distribution Deterioration with Shortages

This article introduces a mathematical model for optimizing total profit by considering various complex factors, including time-dependent shortages, split demand, Gompertz distribution degradation, a linear holding cost function, and partial backlog issues. The two stages of demand, which are time-dependent linear demand and price-dependent exponential demand, occur when there are no shortages. A comprehensive approach to inventory management decision-making is illustrated through a sensitivity analysis and numerical example, highlighting the influence of system limitations on the optimization outcomes. This model is an effective tool for improving inventory management practices and assessing concavity while maximizing profitability. With MATLAB, relationships between model parameters are graphically represented.

Modelling the age distribution of longevity leaders

Human longevity leaders with remarkably long lifespan play a crucial role in the advancement of longevity research. In this paper, we propose a stochastic model to describe the evolution of the age of the oldest person in the world by a Markov process, in which we assume that the births of the individuals follow a Poisson process with increasing intensity, lifespans of individuals are independent and can be characterized by a gamma–Gompertz distribution with time-dependent parameters. We utilize a dataset of the world’s oldest person title holders since 1955, and we compute the maximum likelihood estimate for the parameters iteratively by numerical integration. Based on our preliminary estimates, the model provides a good fit to the data and shows that the age of the oldest person alive increases over time in the future. The estimated parameters enable us to describe the distribution of the age of the record holder process at a future time point.

The Immigrant Mortality Advantage Among Over-65 Middle Eastern and North African Immigrants to the United States.

Immigrants to the United States often have longer life expectancies than their U.S.-born counterparts, however it is unclear whether a similar "immigrant advantage" exists for immigrants from the Middle East and North Africa (MENA). This study uses a novel machine-learning name classifier to offer one of the first national-level examinations of MENA mortality patterns by nativity in the United States. A recurrent neural network model was developed to identify MENA individuals based on given name and surname characteristics. The model was trained on more than 2.5 million mortality-linked social security records in the Berkeley Unified Numident Mortality Database (BUNMD). Mortality rates and life expectancy were estimated using a Gompertz distribution and maximum likelihood estimation, focusing on high-coverage years between 1988 and 2005 and deaths over age 65. Foreign-born MENA men over 65 showed a significant immigrant mortality advantage with a hazard ratio (HR) of 0.64 and an estimated 3.13 additional years of life expectancy at age 65 compared to U.S.-born counterparts. Foreign-born MENA women also exhibited an advantage, with a HR of 0.71 and an additional 2.24years of life expectancy at age 65. This study is one of the first national-level analyses of mortality outcomes among the over-65 MENA population in the United States, finding a MENA immigrant mortality advantage. The results suggest further research is needed to identify and disaggregate the MENA population in health research.

Analysis of the lifetime performance index in Gompertz distribution based on k-record values

The lifetime performance index plays a crucial role in manufacturing procedures as it provides valuable insights into the overall quality of a product. Considering the Gompertz distribution as a widely applicable statistical distribution, this index to emphasize the effect in analyzing lifetime data. This paper investigates the inference techniques associated with the lifetime performance index for the Gompertz distribution when k-record values are available. For this problem, two innovative methods are introduced utilizing the concept of the generalized pivotal variable. These methods are not only applicable to hypothesis tests but also enable the construction of confidence intervals. To further enhance the accuracy of estimation, the study proposes Bayesian estimation techniques based on the Metropolis–Hastings algorithms. To assess the efficacy of these methods, a simulation study is conducted. The results demonstrate that all three procedures exhibit exceptional performance, even when confronted with datasets consisting of small k-record values. Moreover, a dataset is utilized to demonstrate the implementation of these three methods.

Performance of Hypothesis Tests for Gompertz Distribution with Right and Interval Censored Data

Performance of Hypothesis Tests for Gompertz Distribution with Right and Interval Censored Data

Model and Properties of Cauchy Modified Inverse Gompertz Distribution with Application to a Real Data Set

Model and Properties of Cauchy Modified Inverse Gompertz Distribution with Application to a Real Data Set

Social gradient and rural-urban disparities in cancer mortality in Costa Rica

IntroductionData on social inequalities in cancer mortality are sparse, especially in low- and middle-income countries. We aimed to analyze the socioeconomic inequalities in cancer mortality in Costa Rica between 2010 and 2018. MethodsWe linked 9-years of data from the National Electoral Rolls, National Birth Index and National Death Index to classify deaths due to cancer and socioeconomic characteristics of the district of residence, as measured by levels of urbanicity and wealth. We analyzed the fifteen most frequent cancer sites in Costa Rica among the 2.7 million inhabitants aged 20 years and older. We used a parametric survival model based on a Gompertz distribution. ResultsCompared to urban areas, mixed and rural area residents had lower mortality from pancreas, lung, breast, prostate, kidney, and bladder cancers, and higher mortality from stomach cancer. Mortality from stomach, lung and cervical cancer was higher, and mortality from colorectal cancer, non-Hodgkin lymphoma and leukemia was lower in the most disadvantaged districts, compared to the wealthiest ones. ConclusionWe observed marked disparities in cancer mortality in Costa Rica in particular from infection- and lifestyle- related cancers. There are important opportunities to reduce disparities in cancer mortality by targeting cancer prevention, early detection and opportune treatment, mainly in urban and disadvantaged districts.

A generalized Gompertz promotion time cure model and its fitness to cancer data

The cure models based on standard distributions like exponential, Weibull, lognormal, Gompertz, gamma, are often used to analyze survival data from cancer clinical trials with long-term survivors. Sometimes, the data is simple, and the standard cure models fit them very well, however, most often the data are complex and the standard cure models don't fit them reasonably well. In this article, we offer a novel generalized Gompertz promotion time cure model and illustrate its fitness to gastric cancer data by three different methods. The generalized Gompertz distribution is as simple as the generalized Weibull distribution and is not computationally as intensive as the generalized F distribution. One detailed real data application is provided for illustration and comparison purposes.

ON THE MOMENTS OF THE 3-PARAMETER GOMPERTZ DISTRIBUTION

Gompertz distribution is a classical probability distribution extensively used in actuarial science, reliability, and survival analysis. Gompertz distribution also plays a role in various fields, such as biology, economics, and marketing analysis. Some extensions of this distribution have been studied and applied to various problems. In this article, we are concerned with some statistical properties of a 3-parameter Gompertz distribution. This extension of the Gompertz distribution introduced has been used in studying competing risk survival analysis. Our main results are the derivation of moments of this distribution and other statistical properties related to moments, such as moment generating function, mean residual life function, mean inactivity time and Lorenz curve. These results will serve as a complement to the theoretical aspect of the analysis of the distribution.

ANALISIS PREMI TAHUNAN INDIVIDU DENGAN MANFAAT YANG DIBAYARKAN PADA AKHIR TAHUN KEMATIAN

Life insurance is an insurance product that provides guarantees in the form of benefits to the insured's family if the insured dies in the future. The amount of this benefit is influenced by the amount of the annual premium paid by the insurer to the insured. The researcher will calculate the amount of the individual's annual premium on endowment insurance. The researcher will use a fixed interest rate to calculate the discount factor and the Gompertz distribution approach to calculate the insurer's chance of survival. Previously, the researcher will estimate the Gompertz distribution parameters using the Maximum Likelihood Method (MLE) based on the 2011 Indonesian Mortality Table (female). Furthermore, the researcher will calculate the insurer's chance of survival using the Gompertz distribution to obtain the actuarial present value of the initial term annuity and the actuarial present value of the endowment life insurance. In addition, the researcher will calculate the comparison of the present value of the endowment life insurance to the present value of the life annuity. The last step is that the researcher will multiply the comparison of this present value by the benefits received to calculate the amount of the individual's annual premium. The calculation results show that the comparison of the present value of dual-purpose life insurance to the present value of a living annuity is getting smaller as the policy coverage period increases. As a result, the amount of the individual's annual premium will also be smaller. This shows that the amount of the annual premium is influenced by interest rate factors, the insurer's life chances, and the length of the coverage period.